The aim of this paper is to show that it is possible to model the behavior of soils in a large range of strains (10-6 to 10-1), using an elastoplastic law with a unique set of parameters. A multimechanism law with a kinematic hardening was used to model the monotonous and cyclic behavior of sand

Hicher, P. Y.

Kordjani, M.

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics 1995 - Third International Conference on Recent Advances in Geotechnical Earthquake Engineering & Soil Dynamics 04 Apr 1995, 10:30 am - 12:00 pm Modelling of Cyclic Behavior of Sand in Large Range of Strain Amplitudes P. Y. Hicher Laboratory of Mechanics: Soil, Structure, Material, CNRS URA 850, Ecole Centrale de Paris 92290, Chatenay Malabry, France M. Kordjani École Centrale de Paris, France Follow this and additional works at: https://scholarsmine.mst.edu/icrageesd  Part of the Geotechnical Engineering Commons Recommended Citation Hicher, P. Y. and Kordjani, M., "Modelling of Cyclic Behavior of Sand in Large Range of Strain Amplitudes" (1995). International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. 10. https://scholarsmine.mst.edu/icrageesd/03icrageesd/session01/10 This Article - Conference proceedings is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. Proceedings: Third lntemational Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, April 2-7. 1995, Volume I, St. Louis, Missouri Modelling of Cyclic Behavior of Sand in Large Range of Strain Amplitudes Paper No. 1 . 17 P. Y. Hicher and M. Kordjani Laboratory of Mechanics: Soil, Structure, Material, CNRS URA 850, Ecole Centrale de Paris 92290, Chatenay Malabry, France SYNOPSYS The aim of this paper is to show that it is possible to model the behavior of soils in a large range of strains ( w-6 to lQ-1 ), using an elastoplastic law with a unique set of parameters. A multimechanism law with a kinematic hardening was used to model the monotonous and cyclic behaviour of sand. deformations is modeled by an isotropic incremental INTRODUCTION relationship written as follows : In the calculations of important constructions subjected to seismic loading, the effects of non linear and irreversible behaviour of soil on soil-structure interaction have to be considered. Finite element calculations require the use of constitutive models which are able to take into account as closely as possible soil characteristics. We present in this study a method of determining the parameters of a multimechanisms elastoplastic model with kinematic hardening developed at Ecole Centrale de Paris by Hujeux (1985). A database on Hostun sand was used for this purpose, in particular liquefaction tests during cyclic loading. The material tested is Hostun sand (0.1 - 3). It is used by many research laboratories and a large amount of data is available . Its granulometric range caracteristics are: 100 PERC E~FI f!,. i'-- D60= I.IOmm ~50=1.02mm 10 = 0.42 mm c =0.56 I\ 20 " 1E+O 1E-1 GRAIN SIZE ( mm ) MODELISATION The model used in this study was developed at Ecole Centrale de Paris by Hujeux (1985). Its an elastoplastic, multimechanisms model based on the concept of critical state. A kinematic hardening is used to model the cyclic behaviour of the material. Elastic domain The elastic part is isotropic and non linear. The non lineair elastic behavior of soil subjected to very small 43 a,e• = (,Ya)a,crd G=G,,1 (~)· P,., Plastic domain It consists of a multimechanism relationship with a kinematic hardening: Three deviatoric mechanisms, each of them in one plane of the space, and an isotropic purely volumetric mechanism. The state of stress in each plane is represented by Pk and Sk with : (cr .. ) +(cr .. ) P,- Ilk Dk .- 2 2 (eJ. = (eJ. +kJ) • The state strain of the mechanism k is defined by the volumetric plastic strain EC and the deviatoric plastic strain vector ef (ef1 ,ef2 ), and ef2 = 2£~ -Monotonic yield surfaces: We define the yield surface for each deviatoric mechanism: (1) <p is the angle of friction at the critical state. The critical pressure Pc has the following expression: P,; = P"' expfl.E~ where 13 is the plastic compressibility modulus defined in the plane (e,logP) and Pco is the initial critical pressure. The total volumetric stain is the sum of the strains produced 4 by each mechanism: £~ = 2, ( £~) k=l In the deviatoric plane, the yield surfaces are circles which radii grow during the loading, with the hardening variable : (2) The hypothesis of the associated law in a normalized deviatoric plane allows us to define 'l't . . (s*.'l't) \jl~ = a(sm<p- ) (3) pk with (sk, 'II~) : product of the vectors Sk and 'I'~ This expression is the generalized expression of the Roscoe equation. 'I'~ is a function that caracterizes the volumetric plastic flow. 'I'~ is a vector that defines the direction of plastic deviatoric strain. (a) is an expansion factor allowing to modulate Roscoe's equation. Sk pk = Sk2 r The yield surface function of isotropic mechanism is given by ; F4 = I Jil- r4 with P = 7dPc ( 4) This surface can be represented by a segment that has a length of r4m and the direction of the normalized isotropic pressure axis. The rate of the volumetric plastic strain is: d,£~ = A. 4 • 'II~ The hardening variable follows a hyperbolic form: rm = r• +£~ / • • .i'c+eP (5) The development of the hardening variable r4 is given by : (6) 44 -Cyclic yield surfaces: The circles at the representation yield surfaces for the deviatoric mechanisms are tangent to each other point Dk with an exterior normal nk. We obtain: (7) (8) (9) The choice of a function a(fk) allows us to reproduce physical phenomena and limits the domains of the mechanism behavior by regulating the plastic strain amplitude at small deformations : a(rk} = ac +(am -a.}a(r*} (10) if r ~ rhys a(r*)=O for £<10-5 ( r if rhys ~ r ~ rmbl ( } r-rhys for I 05<e< I o-4 ark = rmbl- rhys if rmh1 <r<l a(r.} = l for £>10-4 The fourth mechanism is defined as follows for cyclic loadings: I c I c "th c e E){ f 4c= P-(D4 -n4 r4 )-r4 WI r4 =r4 + • 1, ccyc + Ev (11) SMALL DEFORMATION TESTS Youg modulus E and Poisson's ratio V that caracterize the elastic deformations, and elastic radius rel are determined from triaxial tests at small deformations carried out by El Hosri (1984). These tests where performed using Hostun sand under drained conditions, subjected to an isotropic stress of cr3= 285 kPa and a void ratio e0 ::::: 0.63. We found E::::: 526 MPa v= 0.22 rei= 10-2 1 FJEmax r-... ....... I : I . c';-Fl I I 1 s1mu atwn tgure -axial, strain £ 1 % 0 1E-6 lE-5 lE-4 IIIII I. I I I I~ l expenence r-... i'\. " lE-3 lE-2 Figure 1 The evolution of secant modulus E/E with strain max amplitudes. LARGE DEFORMATION TESTS. Determining the parameters caracterising the critical state of the model M, J3, d, and <1> was accomplished by taking into account the monotonic triaxial test results on loose sand with a void ratio of e0 = 0.73 subjected to isotropic stresses of 100, 200, 400 et 600 kPa (Mohkam 1983). See figures 2, 3, 4 and 5 . Hardening parameters a and b were obtained by optimizing the simulated curves the experimental ones. A particular attention was paid to this operation in order to make the adjustments for the drained and undrained test curves simultaneously. We found a.,.= 0.017 and b = 0.2. Q 100 kPa Q 100kPa ITTT 12 ·.....:: _..,. F" I '. e-. ....:::: 1gure 2 , j'....:.:; ..,.... IJ.'f" 'F ..-6 8 p· I I ~ 1gure ~ F v. ~ ., ~ ~;;iii A' ....:::,. ~ ~ -4 ,. ~ e,% :....._ ....-:: ""' _.. - L -'1 e,% -j 0 14 21 35 0 15 30 drained undrained 4 ~ 8 --2 :::::> ~ ,_.. 0 w e,% £,% 0 14 21 35 0 15 30 Figures 2, 3, 4 and 5 drained and undrained triaxial tests. - experience - simulation 45 CYCLIC TESTS A first attempt to reproduce liquefaction tests results (Mohkam 1983), using the parameters determined above failed. The predicted plastic deformations were generated to fast leading material to rupture. The pore pressure could not , then, increase sufficiently to reproduce liquefaction observed experimentally. We decided then to improve the determination of cyclic parameters, especially the ones regulating the amount of plastic strain at the beginning of each stress reversal. In order to do that we used the results of the cyclic tests carried out in a range of deformation from w-6 to w-2 by (El-Hosri 1984 ). The numerical tests, in the plane (E/Emax,E 1 ), showed, indeed, a large difference with the experimental results, especially in the moderate strains (lQ-4 to 1Q-3), frorn which it could be deduced the necessity to improve this model by refining the parameters values that have a strong influence on the model in that range of strains, i.e rhys• rrnob• a,, and m. The readjustment of these parameters put in evidence a large modification in the model response for monotonic tests under large strains. It was then necessary to reattempt this modelling, keeping into account the behavior of sand at small and large strains. We obtained acyc = 1Q-4,ccyc ::::: 0.04, rmob = 0.9 and m = 0.9. As the model parameters were determined, we proceeded' in simulating the same liquefaction test carried out by Mohkam (1983) on the same material. the test conditions were : eo= 0.73, cr 3;= 200kPa. We obtained this time a very good agreement between simulated and experimental results in the planes: ( Q,E 1 ), ( Q,P) and ( U,E1 ). Figure 6 Figure 7 2 Q 100kPa ~_.... .... ~).:I l; 1 -2 P(100 kPa) 0 0.8 2 Figure 8 0 u -8 L,...__~~._._,~._.__..____.>---.J 1.2 2 4 Figure 10 .--.--~~.......-----.----.--.., 12 El% ; 0 Q 100kPa - ;;; j__;;_~~.........._.........._-...+----4 -8 2 0 -2 ____ U .. L-~ .. :: __ t_- ·:! o --;----~··H--L+ , '' ··[t··t--i -t·1riooti>a·· j 1.-'---..~ .......... ....-......_J.IQ 0 2 Figure 11 10 .J Q 100 kPa J.. , I : , ,-~l~-+LJ : . . . i '-e.% i : -r-·;-~-- 1 +--i i t i !···Y+·· 0 l...i-;...J.... ......... ..W..oi-l.oii..J -10 1.5 0. -1.5 Figures 6, 7, 8, 9, 10, 11 Liquefaction test: evolution of cycique strain amplitude and pore pressure with number of cycles. - experimental results - numerical results. CONCLUSION The increase in pore water pressure during cyclic loadings on saturated sand takes usually place under small strain amplitudes ( typically = 10-'~ <E,< 10-3 ). The determination of soil parameters using classical triaxial tests can lead to unsatisfying results because these tests involve larger strains and the accuracy of strains measurement at the beginning of the loading or during unloading phases is not sufficient. Elastoplastic Hujeux model was constructed so that special attention was given to the progressive development of cyclic plastic strains between 10-6 ( elastic deformations ) and 1 Q-2 · We proposed here a method of determining soil parameters for this model, using, along 46 with classical triaxial test results, (G - y) or ( E - £ 1 ) curves which give more precise informations of the behavior in the range IQ-6 < Ec< IQ-2. A validation of the method on a cyclic triaxial test leading to liquefaction was successfully undertaken. The model was able to reproduce the increase of pore water pressure to complete liquefaction with the right number of cycles and the strain amplitudes were well predicted. NOTATION eij :tensor components strain ei (i=l,2,3) :principal strains sij :tensor components stress si(i= 1 ,2,3) :principal stresses K :elastic bulk modulus G :elastic shear modulus Kref :elastic bulk modulus of reference Gref :elastic shear modulus of reference n :elastic exponent P :effective main pressure Pref :pressure of reference (1 MPa) p =a. +a2 +a3 3 sij = aij-- Poij ad=%( SAj )'~ REFERENCES eij = Eij -- E;, Oij Ed = %r e;1eii ]~ Mohkam, M (1983). Contribution a l'etude experimentale et theorique du comportement des sables sous char_gements cycliques.These I.N.P. Grenoble. EL-Hosri (1984) ."Contribution a l'etude des Rroprietes mecaniques des materiaux",these d'etat Umversite Paris VI. Homsi (1986). "Contribution a l'etude des proprietes mecamques des .sols en Retites deformations a l'essai tnaxial.These de ooctorat de l'E.C.P. Hujeux, J.C (1985) ."Une loi de comportement pour 1e chargement cyclique des sols".Genie Parasismique, presses de l'E.N.P.C. Hieber, P.Y."Comportement mecanique des argiles sur divers chemms de sollicitations monotones et cycliques. Application a une modelisation elastoplastique et viscoplastique."These d'etat Univrsite de Paris VI". Kordjani, M and Hieber, P.Y (1993) Caracterisation et modelisation du comportement gclique des sols seme Seminaire Magllrebin de Genie Parasismique Alger93. Kordjani, M and Hieber, P.Y (1994) Modelisation du comportement cyclique du sable d'Hostun. Rencontre scientifiques de Genie civil, 23,24 mars 1994, xnemes journees de l'A.U.G.C.Toulouse. 

Modelling of Cyclic Behavior of Sand in Large Range of Strain Amplitudes

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Modelling of Cyclic Behavior of Sand in Large Range of Strain Amplitudes

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